What is the value of $\sin(x)$ for the maximum value of $(5+3\sin(x))^2 (7-3\sin(x))^3$.

closed as off-topic by choco_addicted, Brahadeesh, José Carlos Santos, Claude Leibovici, Delta-u Oct 17 at 9:25

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  • 1
    What did you try? Why didn't it work out? Are you confused about any particular facet of solving the problem? – jpmc26 Oct 16 at 22:48

Hint:

AM GM inequality

$$\dfrac{2\cdot3(5+3\sin x)+3\cdot2(7-3\sin x)}{2+3}\ge\sqrt[5]{3^2(5+3\sin x)^22^3(7-3\sin x)^3}$$

The equality occurs if $3(5+3\sin x)=2(7-3\sin x)$

  • Thanks a lot sir, thats what im looking for – Hik Aubergine Oct 16 at 18:39

AM-GM inequality is the neat approach, but if you can't figure it out algebra also helps...

let $z=\sin(x)$, $f(z)=g(z)^2h(z)^3$

$$ f' = 2gg'h^3 + 3gh^2h' = gh^2(2g'h+3hh') $$

since $-1 \le z \le 1$, $g$ and $h$ do not have zeros in this region. Therefore $2g'h=-3hh'$, which will give you the same equation to solve for $z$.

HINT

Let $z = \sin x$. You are asking for what value of $z$ does the function $$f(z) = (5+3z)^2 (7-3z)^3$$ attains its maximum. Taking a derivative would likely help.

UPDATE

As mentioned in the comments below, you are only optimizing over $-1 \le z \le 1$.

  • 1
    Worth mentioning that $x$ is probably meant to be real, so $z$ has to be between $-1$ and $1$. – SmileyCraft Oct 16 at 18:32

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